^V *Jwp5rr*Q&&*?-  I T I  H 


UNIVERSITY  OF  PENNSYLVANIA 


A  GENERALIZED  EQUATION 

OF  THE  VIBRATING  MEMBRANE 

EXPRESSED  IN  CURVILINEAR 

COORDINATES 


BY 


#ARRY  M.   SHOEMAKER 


A  THESIS 

Presented  to  the  Faculty  of  the  Graduate  School  in 

Partial  Fulfillment  of  the  Requirements  for 

the  Degree  of  Doctor  of  Philosophy 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER,  PA. 

1918 


UNIVERSITY  OF  PENNSYLVANIA 

A  GENERALIZED  EQUATION 

OP  THE  VIBRATING  MEMBRANE 

EXPRESSED  IN  CURVILINEAR 

COORDINATES 


BY 

HARRY  M.   SHOEMAKER 


A  THESIS 

Presented  to  the  Faculty  of  the  Graduate  School  in 

Partial  Fulfillment  of  the  Requirements  for 

the  Degree  of  Doctor  of  Philosophy 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER,  PA. 

1918 


QMS* 

S-5 


The  author  wishes  to  express  his  sincere  thanks  to  Professor 
Frederick  H.  Safford,  under  whose  direction  and  supervision 
this  work  was  done. 


CONTENTS. 

PAGE. 

1.  General  Equation  of  the  Vibrating  Membrane 1 

2.  Derivation  and  Discussion  of  the  General  Equations  of 

the  Families  of  Confocal  Conies  used  as  Coordinate 
Systems 3 

3.  Discussion   of  the   General   Equation  of  the   Vibrating 

Membrane 5 

4.  Solution  of  the  Equation  in  Rectangular  and  Circular 

Coordinates 7 

5.  Solution  of  the  Equation  in  Parabolic  Coordinates 9 

6.  Review  of  the  Work  Previously  done  on  the  Equation  in 

Parabolic  and  Elliptic  Coordinates 11 

7.  Bibliography 13 


in 


380309 


1.  GENERAL  EQUATION  OF  THE  VIBRATING 
MEMBRANE. 

The  problem  of  the  vibrating  membrane  has  been  attacked 
by  previous  writers  from  the  standpoint  of  the  shape  of  the 
boundaries  of  the  membrane,  rectangular,  circular,  parabolic, 
elliptic.  No  attempt  has  been  made  to  derive  a  single  equation 
which  contains  all  of  these.  Here  we  shall  derive  such  an  equa- 
tion by  the  use  of  a  general  system  of  curvilinear  coordinates. 
These  coordinates  shall  be  restricted  only  by  the  condition  that 
the  resulting  equation  shall  have  a  solution  in  the  form  of  a  product 
of  functions  of  the  independent  variables  involved,  each  function 
depending  only  upon  a  single  variable. 

If  a  stretched  elastic  membrane  is  fastened  at  the  edges  and  is 
made  to  vibrate  at  right  angles  to  the  plane  of  the  membrane, 
its  equation  of  motion  in  rectangular  coordinates  is, 

dW_    JdW     dW\ 
W  dt2  ~  a  V  dx2  +  dy2  ) 

where  a  is  a  physical  constant  depending  upon  the  mass  per 
unit  area  and  the  tension  along  any  line  in  the  surface  of  the 
membrane.  In  (1)  put  V  =  T-u,  where  T  is  a  function  of  t  only 
and  u  a  function  of  x  and  y  only.  The  equation  then  separates 
into  the  two  parts, 

d2u       d2u  d  T2 

(2)    d?  +  df+K2u=0>     and     w+a2K2T==0' 

where  K  is  any  constant.  Solutions  of  the  latter  are  T  =  sin  aKt 
and  T  =  cos  aKt. 

Now  suppose  x  and  y  are  determined  as  functions  of  two  new 
unrestricted  variables,  x  —  fi(a,  j8),  y  =  /2(a,  0)  and  express 
d2ujdx2  +  d2ujdy2  in  terms  of  these  new  variables, 

dx2^ dy2~  da2l\dx)  +  \dy  )  ]  + dp2l\dx  ) 


THE   VIBRATING  MEMBRANE. 


(3)  +(g)']+m»°.%+£#] 

\vy  J  J      3a3$\_3x    3x      dy    dy  J 
)ot\_dx*+  dy*  \  +  dp  Uz2      dy2]' 


Assume  now  that 
(4) 


*  +  iy  =  /(«  +  i$), 

x  —  iy  =  f(a-  i@). 

This  assumption  will  greatly  simplify  (3).     Putting  this  simpli- 
fied result  in  (2)  we  obtain, 

By  means  of  the  orthogonal  relation 

{dx'J  +  \dy~)  =1"L\^J  +  \te)  J       ^  (^f^Tj^' 
this  equation  becomes  v?o&Cj        1/ctM 

(6)      ^+^__j^rf*Y+r«sLY1  -  .  kVj 

W  d*a  +  dP  Ku\_\da)  +{da)  I  TJ^^ 

The  right  member  of  this  may  be  written  in  a  more  convenient 
form.    From  (4)  we  may  derive  the  relation, 


which  when  put  in  (6)  gives,  £  (of  4  ftf         $^      *ft 


3  u      3  u 

(8)  3rf  +  dj2=-  K2u-f'(a  +  tf)  •/'(«  -  tf).    . 

On  the  assumption  that  this  equation  has  a  solution  of  the  form 
u  =  G(ot)-H((3),  it  is  necessary  that  the  relation  (9)  should  hold. 

(9)  /  '(a  +  W  •/'(«-  #)  =  £(«)  +  TO) 

where  £  and  F  are  functions  which  we  shall  determine  now. 
By  differentiating  first  with  respect  to  a  and  then  with  respect 
to  j3  we  obtain  the  equation 

(10)  /'"(a  +  $)  -/'(a  -  10)  -  f'"(a  -  #)•/'(«  +  #)  =  0. 


^  <3) 


THE   VIBRATING   MEMBRANE.  6 

This  equation  has  the  solutions 

(11)  f'(a  +  ip)  =  B  sin  A(a  +  i@)      and      C  cos  4 (a  +  #), 

A,  B,  C  are  constants. 

These  are  the  functions  then  which  satisfy  the  assumption  in  (9). 
Substituting  in  (9)  we  obtain, 

E(a)+  F(0)  =  B2  sin  A(a  +  ifl  -sin  ^(«  -  i0) 

=  £2(sin2  ^4«  +  sinh2  Aft 
and  (8)  becomes, 

0*11        fi  11 

23+2  +  #2£2w(sinh2  48  +  sin2  ii«)  =  0. 

OCT         OjCT 

Hence  (1)  may  now  be  written, 

a2r       /a2r    e2v\ 

(12)  [£2(sinh2  40  +  sin2  Act)]  -^  =  a2  {  ^  +  g^  J  . 

That  the  assumption  in  (9)  really  has  resulted  in  changing  our 
curvilinear  coordinate  system  to  families  of  orthogonal  confocal 
ellipses  and  hyperbolas  may  be  seen  from  (11)  and  (4).  And  the 
method  here  employed  shows  that  (12)  is  the  most  general 
equation  obtainable  for  the  vibrating  membrane  which  has  for 
its  solution  the  function  V  =  T(t)-G(a)-H(8). 

2.  DERIVATION  AND  DISCUSSION  OF  THE  GENERAL 

EQUATIONS  OF  THE  CONFOCAL  CONICS 

USED  AS  COORDINATE  SYSTEMS. 

From  the  equations  of  the  families  of  the  curves  of  the  orthog- 
onal curvilinear  coordinate  system  in  which  (12)  is  expressed, 
we  purpose  now  to  derive  the  following  systems  as  special  cases, 

(a)  System  of  rectilinear  lines. 

(b)  System  of  concentric  circles  and  radial  lines. 

(c)  System  of  confocal  parabolas. 

In  order  to  do  this,  however,  care  must  be  exercised  in  selecting  a 
general  solution  for  (10).    Take  this  solution  in  the  form 

/'(«  +  i$)  =  B  sin  A(a  +  ip)  -  C  cos  A(a  +  ip). 


4  THE  VIBEATING  MEMBRANE. 

Then 

C  B 

f{a  +  tf)=2  sin  {Aa  +  Aip  +  iD)  +  j  cos  {Aa  +  Aip  +  iD). 

By  using  the  relations  C  =  (L4  sin  B  and  5  =  CA  cos  5  and 
making  various  trigonometric  transformations  and  dropping  the 
dashes  from  the  constants,  we  arrive  at  the  form, 

(13)  x  +  iy  =  C[cos  {Aa  +  B)  cosh  (40  +  0) 

+  t  sin  {Aa  +  5)  sinh  (4/3  +  Z>)] 

in  which  A,  B,  C,  D  are  arbitrary  constants.  By  equating  real 
and  imaginary  parts  and  combining  the  resulting  equations  in 
the  usual  way,  we  obtain 

je2  ifi 

(14)  C2  cosh2  (4/3  +  D)  +  C2  sinh2  (40  +  D)  =  *' 

3u  W2 

(15)  C2  cos2  {Aa  +  5)  ~  C2sin2(4a  +  B)  =  l$ 

as  the  equations  for  the  families  of  the  curves  of  our  coordinate 
system.  It  is  to  be  noted  that  the  focal  distance  of  the  families 
is  the  constant  C. 

(a)  System  of  Rectilinear  Lines. 
Put  C2  sinh2  (4/3  +  D)  =  Jfc2  in  (14). 

(16)  vT&+h=L 

If  C  =  oo  while  k  remains  finite  this  becomes  y  =  ±  k.  The 
relation  between  the  parameter  /3  and  the  new  parameter  A;  is 
k  =  ±  C  sinh  (4/3  +  D)  and  this  will  remain  finite  and  equal 
to  d=  F/3  {F  a  constant),  as  C  =  oo  if  at  the  same  time  4  =  0 
in  such  a  way  that  C-A  =  F,  with  D  =  0. 
Put  C2  cos2  (4a  +  £)  =  A2  in  (15). 

x2  v2 

(17)  ^2  -  (?_  h2  =  !• 

When  C  =  oo  and  ^  remains  finite  this  becomes  x  =  db  h.    The 


THE   VIBKATING  MEMBRANE.  5 

relation  between  the  parameter  a  and  the  new  parameter  h  is 
h  =  ±  C  cos  (Act  +  B)  and  this  will  remain  finite  and  equal 
to  =F  Fa  as  C  =  co  if  B  =  7r/2  and  A  =  0  in  such  a  way  that 
CA  =  F. 

(b)  System  of  Concentric  Circles  and  Radial  Lines. 
Put  C2  cosh2  (Ap  +  D)  =  r2  in  (14)  and  let  C  =  0  while  r  re- 
mains finite,  and  we  obtain 

X2  -\-  y2  =  r2. 

Now  since  r  =  C  cosh  ^4/3  cosh  D  -\-  C  sinh  ^4/3  sinh  Z>,  if  we  let 
D  =  oo  as  C  =  0  in  such  a  way  that  both  C  cosh  D  =  F  and 
C  sinh  D  =  F  then  r2  =  FV  and  equation  (14)  becomes 

(18)  x2+y2=  FV^. 

In  (15)  let  C  =  0  with  5  =  0  and  let  ^4  remain  finite  and  we 
obtain 

(19)  y  =  ±  x  tan  ^4a. 

(c)  System  of  Confocal  Parabolas. 
Put  C2  cosh2  (Aj3  -\-  D)  —  a2  in  (14)  and  change  the  origin  to 
the  left  focus.    Then  let  a  —  C  =  y  and  <r  +  C  =  d.     Now  if 
g  =  co  while  7  remains  finite  (14)  finally  becomes, 

(20)  y2  =  47(7  +  x)         for        7  >  0. 

From  the  equations  connecting  7  with  /3  and  8  with  /3  it  may  be 
shown  that  7  =  A^/2  when  C  =  00  and  D  =  0,  provided  that 
A  =  0  in  such  a  way  that  C--42  =  ^42  and  that  5  =  00  under 
the  same  conditions. 

Put  C2  cos2  (Aa  +  5)  =  <r2  in  (15)  and  proceed  as  in  the  case 
of  (14).     Equation  (15)  then  becomes 

(21)  y2  =  47(7  +  x)        for        7  <  0. 

For  the  same  values  of  the  constants  as  used  in  (20)  together 
with  B  —  0  we  here  obtain  7  =  —  Aa2/2.    Equations  (20)  and 

(21)  become 

(22)  y2  =  2?(A&  +  2x), 

(23)  y2  =  AcPUa2  -  2x). 


6  THE  VIBRATING  MEMBRANE. 

The  system  of  rectilinear  lines  obtained  in  (a)  may  also  be 
derived  from  the  system  of  confocal  parabolas.  Put  AfP  =  p 
in  (22)  and  Ac?  =  q  in  (23)  and  change  the  origin  to  the  point 
(«,  0). 

(24)  f  =  2p(x  +  a)  +  p\ 

(25)  2/2  =  -  2q(x  +  a)  +  q\ 

Now  if  we  let  2pa  =  m?  in  (24)  and  then  let  a  =  oo  in  the 
negative  direction  while  m  remains  finite,  we  obtain 

(26)  y  m  ±  m, 

and  this  condition  can  be  brought  about  by  having  /3  =  0  as 
a  =  oo  in  such  a  way  that  2A02  •  a  =  m2,  for  A  is  finite. 

If  we  let  o  —  q/2  =  —  n  in  (25)  and  then  let  a  =  oo  while  n 
remains  finite, 

(27)  x  =  n. 

And  this  condition  can  be  brought  about  by  having  a  =  oo  as 
a  =  oo  in  such  a  way  that  (^4a2/2)  —  a  always  remains  finite. 
For  those  parabolas  of  the  family  in  (23)  in  which  (Ac?j2)  >  a 
the  parameter  n  is  positive  and  for  those  in  which  (Ao?/2)  <  a 
the  parameter  n  will  be  negative. 

If  we  had  changed  the  origin  of  the  curves  in  (14)  and  (15)  to 
the  right  focus  and  then  sent  the  left  to  infinity  in  the  negative 
direction,  we  would  have  obtained 

y2  =  Ao?(Ao?  +  2x)        and        y*  =  A^(A^  -  2x) 

as  the  equations  of  the  families  of  parabolas.  From  these  the 
families  of  rectilinear  lines  may  be  obtained  by  changing  the 
origin  to  the  point  (—  a,  0)  and  then  sending  the  focus  to  infinity 
in  the  positive  direction. 

3.    DISCUSSION   OF   THE   GENERAL   DIFFERENTIAL 
EQUATION  OF  THE  VIBRATING  MEMBRANE. 

The  same  values  of  the  constants  used  for  the  derivation  of 
the  three  special  systems  of  curvilinear  coordinates  from  the 
general  system  should,  when  put  in  the  general  equation  of  the 


THE   VIBRATING  MEMBRANE. 


vibrating  membrane,  with  the  time  element  removed,  give  three 
special  equations  expressed  in  the  above  mentioned  coordinates. 
With  the  aid  of  (13)  equation  (6)  may  be  written 


(28)     —. j  +  ^  +  tf2CMMcosh2  (A$  +  D) 


d2u       d2u 
da2+dP2 

-  cos2  (Aa  +  B)]  =  0. 
This  separates  into, 


(29)  |^  +  [K2(PA2  cosh2  (A$  +  Z))  -  M2]Y  =  0, 

d2X 

(30)  ^-  -  [K2C?A2  cos2  (Aa  +  B)  -  M2]X  =  0. 

The  family  of  horizontal  lines,  y  =  db  F/3,  were  obtained  from 
the  family  of  confocal  ellipses  by  having  C  =  <x>  and  A  =  0  in 
such  a  manner  that  C-A  =  F  and  Z)  =  0.  If  these  values  are 
put  in  (29),  it  becomes 

d?Y  —  —      M 

(31)  jy+  (K2-  M2)Y  =  0        where        ^  =  y- 

The  family  of  vertical  lines  x  =  =F  Fa  were  obtained  from  the 
family  of  confocal  hyperbolas  by  having  C  =  oo  and  .4  =  0  in 
such  a  way  that  C-A  =  F  and  5  =  tt/2.  If  these  values  are 
put  in  (30)  it  becomes 

J2  y       yr 

(32)  -^  +  M2X  =  0        where        M  =  y . 

And  these  are  the  equations  into  which 

d2u      d2u 

dtf  +  dtf 
separates. 

The  family  of  concentric  circles,  x2  +  y2  =  F2e2AB,  were  ob- 
tained from  the  family  of  confocal  ellipses  by  having  D  =  oo 
and  C  =  0  in  such  a  way  that  C  cosh  D  =  F  and  C  sinh  D  —  F. 
Moreover 

dP2~  Ar  dr>~*~ATdr 


0+^+^=0 


8  THE  VIBRATING  MEMBRANE. 

in  this  instance.     If  these  values  are  put  in  (29)  it  becomes 


(33)  -  -      M 

K=KF        and        M  =  -f. 

A 

The  family  of  radial  lines,  y  =  ±  x  tan  Act,  were  obtained 
from  the  family  of  confocal  hyperbolas  by  having  C  =  0  and 
B  =  0.     If  these  together  with 

<p  =  tan-1  -  =  Aa 

x 

are  put  in  (29)  it  becomes 

(34)  ^f+Jf2X=0        where        ^  =  J- 

And  these  are  the  equations  into  which  the  equation  which  de- 
fines the  function  u  in  circular  coordinates, 


+  -3T  +  353+S,»-0, 


d2u      1  du      1  d2u 

dr2       r  dr       r2  dtp2 
separates. 

It  will  simplify  matters  somewhat,  in  obtaining  the  equations 
in  parabolic  coordinates,  if  we  adopt  the  equation 

ft  11  fl )  11 

(35)  ^  +  ~  +  IW^Msinh2  (48  +  D)  +  sin2  (4«  +  B))  =  0 

in  place  of  (28).    This  separates  into 

(PY 

(36)  ^  +  [#2CM2  sinh2  (A0  +  D)-  M2]Y  =  0, 

(37)  ^  +  [JC2^2  sin2  (Aa  +  B)  +  M2]Z  =  0. 

The  family  of  confocal  parabolas,  y2  =  A{P(AfP  +  2x),  were 
obtained  from  the  family  of  confocal  ellipses  by  having  ^4  =  0 
and  C  =  oo  in  such  a  way  that  C-A2  =  ^4  and  D  =  0.  If  these 
values  are  put  in  (36)  it  becomes 

(PY  - 

(38)  S  +  (*V -  ^/2) F  =  °>        where        *2  "  ^2^2- 


THE   VIBRATING  MEMBRANE.  9 

The  family  of  confocal  parabolas,  y2  =  Ao?{Ac?  —  2x)  were 
obtained  from  the  family  of  confocal  hyperbolas  by  using  the 
same  values  of  the  constants  as  above  and  B  =  0.  If  these  are 
put  in  (37)  it  becomes 

(39)     t4  +  (k2a2  +  M2)X  =  0        where        k2  m  K2-A2. 


4.  SOLUTION  OF  THE  EQUATION  IN  RECTANGULAR 
AND  CIRCULAR  COORDINATES. 

The  complete  solution  of  the  equation  of  a  stretched  elastic 
rectangular  vibrating  membrane  is  obtained  by  an  extension  of 
Fourier's  Theorem  and  may  be  found  in  the  works  of  any  of 
the  writers  on  Partial  Differential  Equations.  The  complete 
solution  representing  the  vibrations  of  a  circular  membrane  is 
obtained  by  the  use  of  Bessel's  Functions.  If  the  membrane 
vibrates  so  that  it  has  circular  symmetry  about  the  axis  per- 
pendicular to  the  plane  of  the  boundary  at  the  origin,  the  Jo 
functions  furnish  the  solution.  If  the  mode  of  vibration  is  a 
function  of  both  the  coordinates  r  and  <p  then  the  Jn  functions 
furnish  the  desired  solution.  And  a  discussion  of  the  problem 
may  be  found  in  any  of  the  works  on  Bessel's  Functions.  What 
we  believe,  however,  to  be  a  contribution  to  the  literature  of 
Bessel's  Functions,  is  a  method  of  obtaining  the  following  funda- 
mental relations  between  these  functions: 

dJn(x)      n 
^      dx     =  x    n^  ~  Jn+lW> 

(l>)    2  —j^-   =   Jn-l(x)   ~  Jn+l(x), 

dJn(x)  n 

(<0   -T-  =  Jn-i(x)  -  -  Jn(x), 

2n 

(d)    ~Jn(x)   =   Jn+l(x)  +  J„-l(x), 


(«) 


dx2 


(^i-1)ViCj^)- 


10  THE   VIBRATING  MEMBRANE. 

The  usual  method  of  obtaining  these  relations  is  to  consider  the 
functions  entirely  independent  of  the  equation  of  which  they 
are  solutions.  Since  the  functions  are  defined  by  a  differential 
equation  it  was  thought  that  the  equation  itself  ought  to  yield 
these  fundamental  relations. 

Starting  with  Jn(x)  as  a  solution  of  Bessel's  Equation, 

and  substituting  z  =  xnv,  we  obtain  v  =  x~nJn(x)  as  a  solution  of 
dh      2w  +  1  dv 

Substituting  this  value  or  v  in  (41) 

(42)  ^  (x-»Jnx)  +  -rr^-r  ^  (x-*Jnx)  +  x-»Jnx  =  0. 

The  corresponding  equation  for  Jn+iX  is 

ff  ,        1T       ,   ,  2ra  +  3  d  , 

(43)  £5  O^'W)  +  -7-  Tx  (x—V^x) 

+  X-n-Vr+lX  =   0. 

Differentiate  (42)  with  respect  to  x,  add  to  (43)  multiplied  by  x 
and  arrange  terms, 

^[ic(x'"J"x)+x~"J^x] 

+ ( x  - 2J^)  •  W*~°J"x) + «"***]-  °- 

A  solution  for  this  is 

(45)  ^  (ar»J«aO  +  af*JWi*  =  0. 

A  corresponding  relation  between  Jnx  and  Jn-\X  may  be 
obtained.    Put  z  =  aT"i>  in  (40) 


THE   VIBRATING   MEMBRANE.  11 

...  dh        2ft  —  1  dv 

dx2  x      dx 

A  solution  for  this  is  v  =  xnJnx.    Put  this  in  (46), 

.._.  d2(xnJnx)      2ft  -  1  d(xnJnx)     . 

(47)  —& x dx—+  xJ»x  =  °- 

The  corresponding  equation  for  Jn-\X  is 

<P(af-y._1g)      2ft  -  3  djx^Jn^x)        n_ 

(48)        dJ X dx +XnlJn-l*=V- 

Differentiate  (47)  with  respect  to  x,  subtract  (48)  multiplied  by 
x  from  it  and  arrange  terms 

*XLt  nr  1 

dx2ldx{-xnJnX>  ~  *V"-i*J 

2ft  -  1  d  T  d  .     _    •  ,       "1 

(49)  "  —^  £  [  & (*"  J"*}  "  xnJn~lX  J 

A  solution  for  this  is 

(50)  -g  (xnJnx)  -  a:nJn_!a;  =  0. 

Relations  (a),  (6),  (c),  (d)  may  be  obtained  from  (45)  and  (50). 
Relation  (e)  may  be  obtained  from  (40)  by  the  substitution, 
z  =  v/  V  x,  where  v  =  V  xJnx. 


5.  SOLUTION  OF  THE  EQUATION  IN  PARABOLIC 
COORDINATES. 

The  equation  of  the  vibrating  membrane  written  in  parabolic 
coordinates  is 

(51)  (Q?+W_=02^__+_j. 

This  separates  into 

d2T 


12  THE   VIBRATING  MEMBRANE. 

and 

(52)  ^-+(JfeV  +  X)X=0, 

d?Y 

(53)  ^r+(W-X)y  =  0. 

Dr.  H.  Weber  in  his  Partiellen  Differentialgleichungen,  Vol.  2, 
page  256,  finds  two  imaginary  solutions  for  (52)  as  limiting  forms 
of  the  hypergeometric  series.  He  also  finds  two  real  solutions 
in  definite  integral  form.  Several  new  solutions  for  special 
values  of  X  will  now  be  obtained. 

Repeated  differentiation  of  (52)  will  point  the  way  to  the 
construction  of  equations  which  when  differentiated  1,  2,  3  •  •  •  n 
times  will  give  (52).  The  nth  equation  of  this  series  can  be 
solved  if  X  has  the  special  value  X  =  ik(2n  +  1),  where  n  is  a 
positive  integer.  And  from  this  solution  a  solution  of  (52)  may 
be  obtained.     It  is 

(54)  X=C^eY^-^n(e-^). 

From  this  a  series  of  functions  depending  upon  n  may  be  ob- 
tained.   For  n  —  0  we  have  the  very  simple  one, 

(55)  X=  C(V«)-iW. 
Weber's  two  solutions  for  X  =  ik  become 


Xx  =  Ci(  V  e)w  •  [  1  +  (-  iko?)  +  t^  (-  ike?) 
(56)  L 


*isi<-*y.+ 


]■ 


(57) 


X2  =  C,(  V  e)"a,V  (-  ike?)  [  1  fj~  (-  ike?) 


+rr|r5(-^)2  +  r^5T7(-^)3---]- 


Equation  (56)  reduces  to  Zi  =  C\(  V  e)-***1  and  this  agrees  with 
(55).  The  series  in  (57)  may  be  summed  by  means  of  the 
integral 


THE  VIBRATING  MEMBRANE.  13 

f  e~x(^x)-1dx  =  2e_IVa;l  1  +  ^3^  +  ^7375^+  •*•], 

which  is  obtained  by  repeated  integration  by  parts  of  the  left 
member.    Applying  this  to  (57)  we  obtain, 

p-ikcfl 

(58)  Z2=  ( V  e) -****(  V-2&)  e**da. 

Jo 

None  of  the  above  results  lend  themselves  to  the  solution  of  the 
problem  of  the  vibrating  membrane  because  they  are  imaginary. 
For  X  =  0,  however,  two  real  solutions  of  (52)  are 

(59)  Xx-aojl      3.4+3.7.4.8      3-7-11-4-8-12"1"  '"  J1 

(60)  kean  -1 


4-8-12-5-9-13 


The  corresponding  y-functions  for  (53)  may  be  found  from  these 
by  replacing  a  with  j8.  These  functions  then  may  be  used  in 
solving  the  problem  of  the  vibrating  membrane  with  parabolic 
boundaries  when  the  mode  of  vibration  is  such  that  X  =  0. 

The  first  two  roots  of  Xx  =  0  and  X2  =  0  in  (59)  and  (60) 
have  been  computed  by  the  writer.    They  are 

,  (*!  =  4.013-  -.,  J:n=5.563..-, 

for  Xt  -  0  j^  _  1Q  246. .  ,t        for  Z2  =  0  j  ^  =  n  ^ ,  % 

The  roots  for  Fi  =  0  and  F2  =  0  are  identical  with  the  corre- 
sponding roots  of  Zi  =  0  and  X%  =  0.  These  roots  are  the 
values  of  the  parameter  which  give  the  nodal  lines  for  this  par- 
ticular mode  of  vibration. 

Weber's  two  imaginary  solutions  for  X  =  0  are, 

X,  =  ( V  e)w  [  1  +  i(-  iko?)  +  |~~  (-  iko?f 
(61)  ,1-3'13        <       V,4  1 


14  THE  VIBRATING  MEMBRANE. 


(62) 


X2  =  ( V  e)ikai  V  ( -  iho?)  [  1  +  ft-  ika2) 

+  peg  ("  *^  +  s^i  ("  ^)3'  •  •]  • 

Equation  (61)  reduces  to  (59),  a  real  solution,  if  a0  =  1.  Equa- 
tion (62)  reduces  to  (60),  a  real  solution,  if  ax  =  V—  i.  These 
results  may  be  obtained  by  performing  the  indicated  multipli- 
cations in  each  case. 

6.  REVIEW  OF  THE  WORK  PREVIOUSLY  DONE  ON 

THE  SOLUTION  OF  THE  EQUATION  IN 

PARABOLIC  AND  ELLIPTIC 

COORDINATES. 

Parabolic  coordinates :  The  real  solutions  of  the  equation 

7)2  x 

— +(Fa2  +  X)X=0 

obtained  by  Dr.  H.  Weber*  and  referred  to  previously  are 

(63)  Xi  -  jf1  s~i(l  -  »)H  cos  [  Vc?(s  *-  *)  +  £  log  —  ]  ds, 

(64)  Xi  =   f  t-»(l  -s)-i  cos  VtfcPis  -h)+  Ij;  log  \ZrA  *- 

The  corresponding  functions  of  /3  may  be  obtained  from  these  by 
changing  a  into  /3  and  X  into  —  X.  Using  these  solutions  Weberf 
discusses  the  vibration  of  regions  bounded  by  parabolic  nodal 
lines.  He  proves  the  theorem,  "  If  a  function  of  u  different  from 
zero,  defined  by  the  equation 

d2u  .  d2u  ,    ., 

is  continuous  and  its  first  differential  quotient  also,  inside  a 
bounded  region  but  vanishes  on  the  boundary,  then  k2  must  be 
real  and  positive."    Such  regions  are  of  three  kinds, 

*  "Partielle  Differentialgleichungen,"  Vol.  2,  page  256. 
t  Mathematische  Annalen,  Vol.  1,  1868. 


THE  VIBRATING  MEMBRANE.  15 

(a)  Bounded  by  two  parabolas  one  from  each  family. 

(b)  Bounded  by  two  parabolas  from  one  family  and  one  from 

the  other. 

(c)  Bounded  by  two  parabolas  from  one  family  and  two  from 

the  other. 
For  (a)  the  boundary  conditions  are  X  =  0  for  a  =  ±  a\  and 
Y  =  0  for  j3  =  ab  ft.  Under  these  conditions  four  particular 
solutions  may  be  found,  u  =  Xx  •  Y\t  u  =  X2-Y2,  u  =  Xi  •  Y2, 
u  =  Xi-Yi.  The  last  two  however  will  not  satisfy  the  condition 
that  u  shall  be  single  valued  within  the  region.  Two  separate 
systems  of  solutions  then  may  be  built  from  u  =  X\  •  Yx  and 
u  =  X2-Y2.  In  each  solution  the  constants  X  and  k  appear  and 
may  be  determined  as  the  roots  of  the  two  transcendental  equa- 
tions involved. 

For  (6)  the  boundary  conditions  are  X  =  0  for  a  =  a\,  a  =  a2 
and  Y  =  0  for  /3  =  ±  ft.  Two  types  of  solutions  will  fit 
these  conditions,  u  =  Yi{AxX\  +  A2X2),  u  =  Y2(AiXi  +  A2X2). 
These  give  rise  to  two  systems  of  solutions.  The  constants  in 
each  solution  are  A\JA2f  k,  X  and  they  may  be  determined  from 
the  three  transcendental  equations  involved. 

For  (c)  the  boundary  conditions  are  X  =  0  for  a  =  «i,  a  =  a2 
and  Y  =  0  for  0  =  ft,  jS  =  &•  One  system  of  solutions  only 
can  be  used  here,  u  =  (^iZi  +  A2X2){BiYx  +  l^l^)  and  there 
are  four  transcendental  equations  to  determine  the  constants 
AxlAt,  B1/Bi,  X,  k. 

M.  Hartenstein*  obtains  solutions  for  the  equation 

d2u      d2u      70 

in  parabolic  coordinates  for  negative  values  of  k2  by  trans- 
forming the  solutions  of  the  equation  in  circular  coordinates 
(Bessel's  Functions)  into  their  corresponding  functions  expressed 
in  parabolic  coordinates.  This  is  a  part  of  a  general  discussion 
of  the  equation  expressed  in  the  four  systems  of  coordinates  used 
in  the  third  part  of  this  paper.  The  results  obtained  contribute 
nothing  to  the  discussion  of  the  problem  of  the  vibrating  mem- 
brane. 

*  Archiv  der  Mathematik  und  Physik  (2),  Vol.  14. 


16  THE  VIBRATING  MEMBRANE. 

Elliptic  coordinates:  The  equation  of  motion  of  the  vibrating 
membrane  expressed  in  elliptic  coordinates  is, 

dW  fdW      dW\ 

(65)  (cosh2  0  -  cos2  a)  -^  =  a2  ^  +  ^  J 

and  separates  into 

¥+«  =  » 

and 

S  u       d  u 

(66)  ^  +  ^  +  F(cosh2  0  -  cos2  a)u  =  0. 

This  last  equation  separates  into, 

(67)  -^  -  (P  cos2  a  -  \)X  =  0, 

(68)  ^  +  (fc2  cosh2 18  -  X)  Y  =  0. 

Dr.  Heine*  shows  that  (66)  is  a  limiting  form  of  the  differential 
equation  of  the  Lame  Functions  in  the  same  way  as  the  differ- 
ential equation  of  the  Bessel's  Function  is  a  limiting  form  of 
the  differential  equation  of  the  Spherical  Functions.  He  solves 
(67)  by  assuming  that  the  constant  X  depends  in  a  definite  way 
(as  a  root  of  a  known  transcendental  equation)  on  the  constant  k 
and  the  solutions  periodic  functions  of  a.  Four  classes  of  func- 
tions are  found  to  be  solutions  of  (67)  under  these  assumptions. 
They  are, 

(a)  Series  of  cosines  in  ascending  even  multiples  of  a. 

(b)  Series  of  cosines  in  ascending  odd  multiples  of  a. 

(c)  Series  of  sines  in  ascending  odd  multiples  of  a. 

(d)  Series  of  sines  in  ascending  even  multiples  of  a. 

E.  Mathieut  in  discussing  the  vibratory  movement  of  an 
elliptic  membrane,  uses  periodic  functions  as  solutions  also  of 
the  above  equations.  He  shows  that  the  general  solution  of 
such  linear  differential  equations  of  the  second  order  as  (67) 
and  (68)  is  composed  of  two  parts,  one  zero  for  the  zero  value  of 
the  argument  and  the  other  a  maximum.    If  the  constant  X  has 

*  "Handbuch  der  Kuglefunctionen,"  page  401. 
t  Journal  de  LiouvUle  (2),  Vol.  13,  1886. 


THE  VIBKATING  MEMBRANE.  17 

the  form  X  =  g2  +  bkA  +  ck*  -f-  •  *  •  where  g  is  an  integer  and 
b,  c,  etc.  constants  determined  by  the  periodic  property,  the 
solutions  of  (67)  and  (68)  may  be  written  as  periodic  functions 
of  a  and  |8  in  ascending  powers  of  k2,  and  they  become  zero  g 
times  from  0  to  w.  For  the  special  case  when  k  =  0  these  solu- 
tions become  Xi  =  sin  (ga),  X%  =  cos  (ga),  Yi  =  sin  (gi(3), 
F2  =  cos  (gift).  Solutions  for  these  equations  are  also  given  in 
series  according  to  the  ascending  powers  of  sin  and  cos  of  a 
and  /3.  In  this  case  the  general  solution  of  (67)  is  the  sum  of 
solutions  one  even  and  the  other  odd  in  sin  a  and  one  even  and 
the  other  odd  in  cos  a.  These  results  agree  with  those  given 
by  Heine.  From  these  solutions  two  types  of  vibration  are 
possible.  And  some  simple  cases  of  hyperbolic  nodal  lines  are, 
the  major  axis  alone,  both  together,  two  asymptotes  of  the  same 
hyperbola,  two  asymptotes  of  the  same  hyperbola  with  the  major 
and  minor  axes.  The  elliptic  nodal  lines  are  obtained  from 
Fi  =  0  and  F2  =  0  as  follows:  If  /3  =  B  on  the  boundary  of 
the  membrane  when  Y  =  0,  the  constant  X  may  be  found  from 
Y(B,  X)  =  0.  If  Xi,  X2,  X3,  •  •  •  are  the  roots  of  this  equation  in 
order  of  increasing  magnitude,  F(j3,  Xn)  =  0  will  give,  through 
its  roots  in  /3,  the  parameters  of  the  elliptic  nodal  lines.  This 
equation  is  shown  to  have  n  —  1  roots  less  than  B,  and  there- 
fore the  number  of  elliptic  nodal  lines  is  n  —  1. 

F.  Pockels*  has  investigated  the  function  defined  by  (66)  for 
the  region  bounded  by  any  two  ellipses,  j3  =  j3i  and  j3  =  jff2  and 
any  two  hyperbolas,  a  =  ai  and  a  =  a2  under  the  supposition 
that  «  =  0on  all  four  sides.  He  proves  the  following  theorem: 
"  There  is  for  such  a  region  a  doubly  infinite  series  of  normal 
functions  which  satisfy  the  boundary  condition  u  =  0  on  all 
four  sides,  and  of  which  a  definite  number  (m  —  1)  of  elliptic 
and  a  definite  number  (n  —  1)  of  hyperbolic  nodal  lines  are 
characteristic,  (m  =  1,  2,  •  •  •,  00,  n  =  1,  2,  •  •  •,  00)."  By  cut- 
ting the  bounded  region  along  the  line  connecting  the  foci,  and 
assuming  that  u  and  its  first  derivative  are  continuous  along  this 
new  boundary,  the  new  region  thus  formed  may  be  regarded  as  a 
simply  connected  Riemann  surface.  This  makes  possible  the 
consideration  of  regions  bounded  by  less  than  four  curves,  as 

*  "  Ueber  die  Partielle  Differentialgleichung  Au  +  &2w  =  0." 


18  THE  VIBRATING  MEMBRANE. 

for  example,  those  bounded  by  two  ellipses  and  two  pieces  of 
the  same  hyperbola,  two  hyperbolas  and  one  ellipse,  etc. 

Hartenstein*  has  solved  (66)  for  ¥  negative.  His  method  is 
entirely  different  from  that  employed  by  the  above  writers  and 
consists  in  extending  Bessel's  Functions  by  transformations 
which  will  make  them  solutions  of  (66).  The  transformations 
are  many  and  complex  and  the  results  add  nothing  to  what  has 
already  been  noted. 

Dr.  F.  Lindemannf  solves  (67)  and  (68),  obtaining  non- 
periodic  functions  of  a  and  /3  for  his  solutions.  He  finds  four 
special  solutions  for  each  in  series  form  which  include  those 
periodic  solutions  of  Heine  noted  above,  as  special  cases. 

7.  BIBLIOGRAPHY. 

E.  H.  Barton.    Textbook  on  Sound. 

W.  E.  Byerly.    Fourier's  Series  and  Spherical,  Cylindrical  and 

Ellipsoidal  Harmonics. 
Grey  and  Mathews.    Treatise  on  Bessel  Functions. 

E.  Heine.    Handbuch  der  Kugelfunctionen. 

H.  Hartenstein.    Archiv  der  Mathematik  und  Physik,  Vol.  14, 
1890. 

Dissertation,  Leipzig,  1887. 
G.  Lame.    Coordonnees  Curvilignes. 

Lecons  sur  Elasticite. 

F.  Lindemann.     Mathematische  Annalen,  Vol.  22,  1883. 

E.  Mathieu.    Journal  de  Liouville,  Vol.  13,  1868. 
Cours  de  Physique  Mathematique,  1873. 

Michell.    Messenger  of  Mathematics,  Vol.  19,  1890. 

F.  Pockels.    Ueber    die    Partielle    Differentialgleichung   Am 
+  Wu  =  0. 

Lord  Rayleigh.    Theory  of  Sound,  Vol.  1. 

B.  Riemann.    Partiellen  Differentialgleichungen  und  deren  An- 

wendung  auf  Physikalische  Fragen. 
E.  Reinstein.    Annalen  der  Physik,  Vol.  35,  1911. 
H.  Weber.    Die  Partiellen  Differentialgleichungen,  Vol.  2. 

Mathematische  Annalen,  Vol.  1. 
A.  Winkelmann.    Handbuch  de  Physik,  Vol.  2. 

*  Archiv  der  Mathematik  und  Physik  (2),  Vol.  14. 
t  Math.  Annalen,  Vol.  22,  1883. 


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